Finally, we evaluated the long-term stability of the PRTR locking. We recorded the PRTR of the open loop and closed loop for 10,000 s at a sampling rate of 0.01 s as well as the reference signal. The recorded results are shown in Fig. REF  (a), (b) and (c), respectively.
During the whole measurement, the closed-loop PRTR and the reference signal have a standard deviation of \(1.86\times 10^{-3}\)  and \(4.34\times 10^{-4}\) , while that of the open-loop is \(1.13\times 10^{-1}\) , two orders of magnitude higher than the closed-loop.
The Allan deviations of the above recorded data are shown in Fig. REF  (d). In the closed loop regime, the Allan deviation is improved from \(1.14\times 10^{-3}\)  to \(1.32\times 10^{-4}\)  at 1 s averaging time, one order of magnitude better than that of the open loop regime increasing as a result of the drift over time. It is worth noting that the lowest Allan deviation of the closed loop reaches \(4.39\times 10^{-5}\)  at 1000 s averaging time, nearly three orders of magnitude lower than the open loop regime. As a benchmark, the reference signal reaches its lowest Allan deviation of \(1.13\times 10^{-5}\)  at 100 s averaging time, which ensures long-term stability of the locking system.
Compared to Ref. [1]}, our result with respect to the Allan deviation of PRTR is one order of magnitude better. Supposing an interrogation time of 50 ms for a compact and transportable AI, we estimate that the error introduced by the PRTR in the g-measurement would be reduced from \(10^{-7}\)  g to \(10^{-10}\)  g by calculation. This improvement may suppress the effect of the PRTR instability to a level negligible to a compact and high-precision AI.
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